Onboard Exact Solution to the Full-Body Relative Orbital Motion Problem D. Condurache * and A. Burlacu † Gheorghe Asachi Technical University of Iasi, 700050 Iasi, Romania DOI: 10.2514/1.G000316 This paper provides a representation theorem of the onboard exact solution for the six-degree-of-freedom relative orbital motion problem. This problem is quite important, due to its numerous applications: spacecraft formation flying, rendezvous operations, autonomous missions. The common approach is to consider the relative translation and rotation dynamics for the leader–deputy spacecraft formation to be modeled using vector-based formalisms. The proposed approach makes use of a complete dual orthogonal tensor-based construction, which allows for a new description of the full-body relative orbital motion problem. The approach chosen for representing the solution is very short in comparison with the previous reports on the same problem and gives a new formulation that could give important computational benefits. Nomenclature A = real tensor A = dual tensor a = real number a = real vector a = dual number a = dual vector f c = true anomaly h c = specific angular momentum of the leader satellite L V 3 ; V 3  = dual-tensor set p c = conic parameter ^ q = real quaternion ^ q = dual quaternion R = real numbers set R = dual numbers set SO 3 = orthogonal real tensors set SO 3 = orthogonal dual-tensor set SO R 3 = time depending real tensorial functions SO R 3 = time depending dual tensorial functions U = unit dual quaternions set U R = time depending unit dual quaternions functions V 3 = real vectors set V 3 = dual vectors set V R 3 = time depending real vectorial functions V R 3 = time depending dual vectorial functions I. Introduction T HE analysis of relative motion began in the early 1960s with the paper of Clohessy and Wiltshire [1], who obtained the equations that model the relative motion in the situation in which the chief spacecraft has a circular orbit and the attraction force is not affected by the Earth oblateness. They linearized the nonlinear initial value problem that models the relative motion by assuming that the relative distance between the two spacecraft remains small during the mission. The Clohessy–Wiltshire equations are still used today in rendezvous maneuvers, but they cannot offer a long-term accuracy because of the secular terms present in the expression of the relative position vector. Independently, Lawden [2], Tschauner and Hempel [3], and Tschauner [4] obtained the solution to the linearized equations of motion in the situation in which the chief orbit is elliptic, but their solutions still involved secular terms and had singularities. The singularities in the Tschauner–Hempel equations were removed first by Carter [5] as well as Yamanaka and Andersen [6]. Formation flying is a highly complex research topic. One of the main problems is the design of equations for the relative motion with a long-term accuracy degree raised, together with the need to obtain a more accurate solution to the relative orbital motion problem [7]. Gim and Alfriend [8] used the state transition matrix in the study of the relative motion. The main goal was to express the linearized equations of motion with respect to the initial conditions, with appli- cations in formation initialization and reconfiguration. Different research for more accurate equations of motion starting from the nonlinear initial value problem that models the motion was reported Gurfil and Kasdin [9], who derived the closed-form expression of the relative position vector, but only when the reference trajectory is circular. Similar expressions for the law of relative motion starting from the nonlinear model are presented in [7,10–12]. The relative orbital motion problem was also studied from the point of view of the associated differential manifold. Gurfil and Kholshevnikov [13] introduced a metric that helps to study the relative distance between Keplerian orbits. Gronchi [14,15] also introduced a metric between two confocal Keplerian orbits and used this instrument in problems of asteroid and comet collisions. Using an approach published in 1995 [16] by one of the authors of this paper, in 2007 Condurache and Martinusi [17,18] offered the closed-form solution to the nonlinear unperturbed model of the relative orbital motion. The method led to closed-form vectorial coordinate- free expressions for the relative law of motion and relative velocity. As the main contribution, the method involves the Lie group of proper orthogonal tensor functions and its associated Lie algebra of skew- symmetric tensor functions. Then, the solution was generalized to the problem of the relative motion in a central force field [19–21]. An inedite solution to the Kepler problem by using the algebra of hypercomplex numbers was offered in [22]. Based on this solution and by using the hypercomplex eccentric anomaly, a unified closed-form solution to the relative orbital motion was determined [23]. The relative motion between the leader and the deputy is a six-degree- of-freedom (6-DOF) motion, which represents the coupling of the relative translational motion with the rotational one. In recent years, increasing attention has been paid to the modeling of the 6-DOF motion of spacecraft [24–26]. Also, controlling the relative pose of satellite formation is a very important research subject [27,28]. The common approach is to consider the relative translational and rotational dynamics Presented as Paper 2014-4363 at the AIAA/AAS Astrodynamics Specialist Conference, San Diego, CA, 4–7 August 2014; received 30 November 2015; revision received 12 June 2016; accepted for publication 19 June 2016; published online 7 September 2016. Copyright © 2016 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal and internal use, on condition that the copier pay the per-copy fee to the Copyright Clearance Center (CCC). All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0731- 5090 (print) or 1533-3884 (online) to initiate your request. *Professor, Department of Theoretical Mechanics, D. Mangeron Street No. 59. Senior Member AIAA. † Associate Professor, Department of Automatic Control and Applied Informatics, D. Mangeron Street No. 59. 2638 JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 39, No. 12, December 2016 Downloaded by Daniel Condurache on November 24, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.G000316